3.10.50 \(\int \frac {(4 b+a x^3) (-b-a x^3+x^4)}{x^4 \sqrt [4]{b+a x^3} (-b-a x^3+2 x^4)} \, dx\)

Optimal. Leaf size=72 \[ -\frac {4 \left (a x^3+b\right )^{3/4}}{3 x^3}-2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{2} x}\right )+2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{a x^3+b}}\right ) \]

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Rubi [F]  time = 3.45, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4*b + a*x^3)*(-b - a*x^3 + x^4))/(x^4*(b + a*x^3)^(1/4)*(-b - a*x^3 + 2*x^4)),x]

[Out]

(-4*(b + a*x^3)^(3/4))/(3*x^3) + 4*b*Defer[Int][1/((b + a*x^3)^(1/4)*(b + a*x^3 - 2*x^4)), x] + a*Defer[Int][x
^3/((b + a*x^3)^(1/4)*(b + a*x^3 - 2*x^4)), x]

Rubi steps

\begin {align*} \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx &=\int \left (\frac {4 b}{x^4 \sqrt [4]{b+a x^3}}+\frac {a}{x \sqrt [4]{b+a x^3}}+\frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )}\right ) \, dx\\ &=a \int \frac {1}{x \sqrt [4]{b+a x^3}} \, dx+(4 b) \int \frac {1}{x^4 \sqrt [4]{b+a x^3}} \, dx+\int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^3\right )+\frac {1}{3} (4 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{b+a x}} \, dx,x,x^3\right )+\int \left (\frac {4 b}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )}+\frac {a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )}\right ) \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^3}\right )-\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^3\right )+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^3}\right )-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )+\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}+\frac {2 a \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4*b + a*x^3)*(-b - a*x^3 + x^4))/(x^4*(b + a*x^3)^(1/4)*(-b - a*x^3 + 2*x^4)),x]

[Out]

Integrate[((4*b + a*x^3)*(-b - a*x^3 + x^4))/(x^4*(b + a*x^3)^(1/4)*(-b - a*x^3 + 2*x^4)), x]

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IntegrateAlgebraic [A]  time = 1.91, size = 72, normalized size = 1.00 \begin {gather*} -\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}-2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{2} x}\right )+2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{b+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((4*b + a*x^3)*(-b - a*x^3 + x^4))/(x^4*(b + a*x^3)^(1/4)*(-b - a*x^3 + 2*x^4)),x]

[Out]

(-4*(b + a*x^3)^(3/4))/(3*x^3) - 2^(3/4)*ArcTan[(b + a*x^3)^(1/4)/(2^(1/4)*x)] + 2^(3/4)*ArcTanh[(2^(1/4)*x)/(
b + a*x^3)^(1/4)]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} - x^{4} + b\right )} {\left (a x^{3} + 4 \, b\right )}}{{\left (a x^{3} - 2 \, x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b),x, algorithm="giac")

[Out]

integrate((a*x^3 - x^4 + b)*(a*x^3 + 4*b)/((a*x^3 - 2*x^4 + b)*(a*x^3 + b)^(1/4)*x^4), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}+4 b \right ) \left (-a \,x^{3}+x^{4}-b \right )}{x^{4} \left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (-a \,x^{3}+2 x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b),x)

[Out]

int((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} - x^{4} + b\right )} {\left (a x^{3} + 4 \, b\right )}}{{\left (a x^{3} - 2 \, x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b),x, algorithm="maxima")

[Out]

integrate((a*x^3 - x^4 + b)*(a*x^3 + 4*b)/((a*x^3 - 2*x^4 + b)*(a*x^3 + b)^(1/4)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a\,x^3+4\,b\right )\,\left (-x^4+a\,x^3+b\right )}{x^4\,{\left (a\,x^3+b\right )}^{1/4}\,\left (-2\,x^4+a\,x^3+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*b + a*x^3)*(b + a*x^3 - x^4))/(x^4*(b + a*x^3)^(1/4)*(b + a*x^3 - 2*x^4)),x)

[Out]

int(((4*b + a*x^3)*(b + a*x^3 - x^4))/(x^4*(b + a*x^3)^(1/4)*(b + a*x^3 - 2*x^4)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3+4*b)*(-a*x**3+x**4-b)/x**4/(a*x**3+b)**(1/4)/(-a*x**3+2*x**4-b),x)

[Out]

Timed out

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